Simplify; express your answer in exponential form. Assume $t\neq 0, q\neq 0$. $\dfrac{{(t^{4}q^{-3})^{-3}}}{{(t^{5}q^{-2})^{2}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(t^{4}q^{-3})^{-3} = (t^{4})^{-3}(q^{-3})^{-3}}$ On the left, we have ${t^{4}}$ to the exponent ${-3}$ . Now ${4 \times -3 = -12}$ , so ${(t^{4})^{-3} = t^{-12}}$ Apply the ideas above to simplify the equation. $\dfrac{{(t^{4}q^{-3})^{-3}}}{{(t^{5}q^{-2})^{2}}} = \dfrac{{t^{-12}q^{9}}}{{t^{10}q^{-4}}}$ Break up the equation by variable and simplify. $\dfrac{{t^{-12}q^{9}}}{{t^{10}q^{-4}}} = \dfrac{{t^{-12}}}{{t^{10}}} \cdot \dfrac{{q^{9}}}{{q^{-4}}} = t^{{-12} - {10}} \cdot q^{{9} - {(-4)}} = t^{-22}q^{13}$